Abstract
In this article, let $\Sigma\subset\mathbf{R}^{2n}$ be a compactconvex Hamiltonian energy surface which is symmetric with respect tothe origin, where $n\ge 2$. We prove that there exist at least twogeometrically distinct symmetric closed trajectories of the Reebvector field on $\Sigma$.
Highlights
Introduction and main resultsIn this article, let Σ be a fixed C3 compact convex hypersurface in R2n, i.e., Σ is the boundary of a compact and strictly convex region U in R2n
We denote the set of all compact convex hypersurfaces which are symmetric with respect to the origin by SH(2n), i.e., Σ = −Σ for Σ ∈ SH(2n)
There is a long standing conjecture on the number of closed characteristics on compact convex hypersurfaces in R2n:
Summary
Let Σ be a fixed C3 compact convex hypersurface in R2n, i.e., Σ is the boundary of a compact and strictly convex region U in R2n. We denote the set of all such hypersurfaces by H(2n). We denote the set of all compact convex hypersurfaces which are symmetric with respect to the origin by SH(2n), i.e.,. T (Σ) the set of geometrically distinct closed characteristics (τ, y) on Σ. A closed characteristic (τ, y) on Σ ∈ SH(2n) is symmetric if {y(R)} = {−y(R)}, non-symmetric if {y(R)} ∩ {−y(R)} = ∅ It was proved in [LLZ] that a prime characteristic (τ, y) on Σ ∈ SH(2n) is symmetric if and only if y(t). There is a long standing conjecture on the number of closed characteristics on compact convex hypersurfaces in R2n:. Where Ts(Σ) denotes the set of geometrically distinct symmetric closed characteristics (τ, y) on Σ. Let N, N0, Z, Q, R, and C denote the sets of natural integers, non-negative integers, integers, rational numbers, real numbers, and complex numbers respectively. In this article we use only Q-coefficients for all homological modules
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